"Complementary and supplementary angles" is the much required stuff for the students who study math in school level.

Let us have a clear understanding about complementary angles and supplementary angles.

**Complementary Angles :**

**If the sum of two angles is 90⁰, then those two angles are called as complementary angles.**

**Example :**

30° and 60° are complementary angles.

Because 30° + 60° = 90°.

Clearly, 30° is the complement of 60° and 60° is the complement of 30°.**Supplementary Angles :**

**If the sum of two angles is 180⁰, then those two angles are called as supplementary angles.**

120° and 60° are supplementary angles.

Because 120° + 60° = 180°.

Clearly, 120° is the supplement of 60° and 60° is the supplement of 120°.

Let us see, how the stuff "complementary and supplementary angles" appears on picture.

To have better understanding, let us do some problems on "Complementary and supplementary angles".

**Example 1 :**

The measure of an angle is 41°. What is the measure of a complementary angle?

**Solution :**

Let "x" be the measure of a complementary angle required.

Since "x" and 41° are complementary angles, we have

x + 41° = 90°

x = 90° - 41°

x = 49°

**Hence the measure of the complementary angle is 49°**

**Example 2 :**

The measure of an angle is 62°. What is the measure of a complementary angle?

**Solution :**

Let "x" be the measure of a complementary angle required.

Since "x" and 62° are complementary angles, we have

x + 62° = 90°

x = 90° - 62°

x = 28°

**Hence the measure of the complementary angle is 28°**

**Example 3 :**

The measure of an angle is 108°. What is the measure of a supplementary angle?

**Solution :**

Let "x" be the measure of a supplementary angle required.

Since "x" and 108° are supplementary angles, we have

x + 108° = 180°

x = 180° - 108°

x = 72°

**Hence the measure of the supplementary angle is 72°**

**Example 4 :**

The measure of an angle is 89°. What is the measure of a supplementary angle?

**Solution :**

Let "x" be the measure of a supplementary angle required.

Since "x" and 41° are supplementary angles, we have

x + 89° = 180°

x = 180° - 89°

x = 91°

**Hence the measure of the supplementary angle is 91°**

**Example 5 :**

Two angles are complementary. If one of the angles is double the other angle, find the two angles.

**Solution :**

Let "x" be one of the angles.

Then the other angle = "2x"

Since "x" and "2x" are complementary angles, we have

x + 2x = 90°

3x = 90°

x = 30° and 2x = 60°

**Hence the two angles are 30° and 60°**

**Example 6 :**

Two angles are complementary. If one angle is two times the sum of other angle and 3, find the two angles.

**Solution :**

Let "x" and "y" be the two angles which are complementary.

So, we have x + y = 90° --------> (1)

From the information, "one angle is two times the sum of other angle and 3", we have

x = 2(y+3)

x = 2y + 6 ------->(2)

Now plug x = 2y + 6 in equation (2)

(1)-------> 2y + 6 + y = 90

3y + 6 = 90

3y = 84

y = 28

Now, plug y = 28 in equation (2).

(2) --------> x = 2(28) + 6

x = 56 + 6

x = 62

**Hence the two angles are 62° and 28° **

**Example 7 :**

The measure of an angle is 3/4 of 60°. What is the measure of the complementary angle ?

**Solution :**

Let "x" be the measure of a complementary angle required.

Given angle = 3/4 of 60° = (3/4)x60° = 3x15° = 45°

Since "x" and 45° are complementary angles, we have

x + 45° = 90°

x = 90° - 45°

x = 45°

**Hence the measure of the complementary angle is 45°**

**Example 8 :**

Two angles are supplementary. If one angle is 36° less than twice of the other angle, find the two angles.

**Solution :**

Let "x" and "y" be the two angles which are supplementary.

So, we have x + y = 180° ----------->(1)

From the information, "one angle is 36

From the information, "one angle is 36° less than twice of the other angle", we have

x = 2y - 36 ----------->(2)

Now plug x = 2y - 36 in equation (1)

(1)-------> 2y - 36 + y = 180

3y - 36 = 180

3y = 216

y = 72

Now, plug y = 72 in equation (2).

(2) --------> x = 2(72) - 36

x = 144 - 36

x = 108

**Hence the two angles are 108° and 72° **

**Example 9 :**

An angle and its one-half are complementary. Find the angle.

**Solution :**

Let "x" be the required angle.

Its one half is x/2

Since "x" and "x/2" are complementary, we have

x + x/2 = 90°

(2x + x)/2 = 90°

3x/2 = 90°

3x = 180°

x = 60°

**Hence the required angle is 60°**

**Example 10 :**

Two angles are supplementary. If 5 times of one angle is 10 times of the other angle. Find the two angles.

**Solution :**

Let "x" and "y" be the two angles which are supplementary.

So, we have x + y = 180° -------->(1)

From the information, "5 times of one angle is 10 times of the other angle", we have

5x = 10y =====> x = 2y ---------(2)

Plug x = 2y in equation (1)

2y + y = 180

3y = 180

y = 60

Plug y = 60 in equation (2)

x = 2(60)

x = 120

**Hence the two angles are 60° and 120****°.**

So far we have seen problems on "complementary and supplementary angles "without pictures

Apart from the above examples, let us do some problems on "Complementary and supplementary angles" with pictures.

**Example 11 :**

Find the value of "x" in the figure given below.

**Solution :**

From the picture above, it is very clear that the angles "x" and "2x" are complementary.

So, we have x + 2x = 90°

3x = 90°

x = 30°

**Hence the value of "x" is 30°**

**Example 12 :**

Find the value of "x" in the figure given below.

**Solution :**

From the picture above, it is very clear that the angles (x+1), (x-1) and (x+3) are complementary.

So, we have (x+1) + (x-1) + (x+3) = 90

3x + 3 = 90

3x = 87

x = 29

**Hence the value of "x" is 29**

**Example 13 :**

Find the value of "x" in the figure given below.

**Solution :**

From the picture above, it is very clear that (2x+3) and (x-6) are supplementary angles.

So, we have (2x+3) + (x-6) = 180°

2x + 3 + x - 6 = 180°

3x - 3 = 180

3x = 183

x = 61

**Hence the value of "x" is 61.**

**Example 14 :**

Find the value of "x" in the figure given below.

**Solution :**

From the picture above, it is very clear (5x+4), (x-2) and (3x+7) are supplementary angles.

So, we have (5x+4) + (x-2) + (3x+7) = 180°

5x + 4 + x -2 + 3x + 7 = 180°

9x + 9 = 180

9x = 171

x = 19

**Hence the value of "x" is 19**

After having gone through the stuff and examples on "Complementary and supplementary angles", we hope that the students would have understood the stuff "Complementary and supplementary angles".

**To know more about complementary and supplementary angles, please click here**

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